The solution to the boundary value problem (REF ) - () is given by [1]} for the case \( P_0 = 0 \) , \( P_1 \ne 0 \) , and, in the case \( P_0 \ne 0 \) , \( P_1 = 0 \) , by the solution to axisymmetric flow through a pipe. Thus, we have
\(w_0\left(r,z,t\right) = \frac{1}{4}P_0\left(r^2 - 1\right) + \frac{i P_1}{\alpha ^2}\left(1 - \frac{J_0\left(i^{3/2}\alpha r\right)}{J_0\left(i^{3/2}\alpha \right)}\right)e^{it},\) 
